Analysis of Variance
A model introduces the population mean for each level . Here the parameter is the overall average, and is known as the ith factor effect. The hypothesis testing problem to detect ``some effects'' of factor level becomes
versus for some .
Equivalently we can write the hypothesis testing problem as follows:
versus for some .
The data from k groups are arranged either (a) all in a single variable with another categorical variable indicating factor levels, or (b) in multiple columns each of whose variables represents a factor level. The statistical inference begins with calculation of the sample mean within group for every factor level , which is the point estimate of . It is also useful to obtain the sample standard deviation within the group, that is, the square root of .
We then proceed to compute the analysis of variance table (AOV table) which summarizes the degree of freedom (df), the sum of squares (SS), and mean squares (MS).

is the sum of squares between groups,
having
degrees of freedom.
Thus, the mean sqaure is given by

is the sum of squares within groups,
having
degrees of freedom.
Thus, the mean sqaure is given by

is the total sum of squares,
having
degrees of freedom.
It can be decomposed into
Here
Under the null hypothesis the test statistic